On the book Algebraic curves and Riemann Surface by Rick Miranda, page 169, I see the following definition:
(Last part of definition 1.1) A complex Riemann surface $X$ is an algebraic curve if the space $\mathcal M(X)$ of global meromorphic functions separates the points and tangents of $X.$
Of course, usually, “algebraic curves” means the vanishing set of a polynomial in two variables. So how is the definition above related to the usual meaning of an algebraic curve?
It takes an argument to show that this condition : $\mathcal{M}(X)$ separates points and tangents is sufficient to guarantee that $X$ admits a holomorphic embedding into a complex projective space $\mathbb{P}^n$, $i:X\to \mathbb{P}^n$. Once you have that, Chow's Theorem implies that $X$ is in fact defined by algebraic equations. In particular, this shows that $X$ admits a structure of a projective algebraic variety.
The main theorem that one needs here is :
The content of this is that there are "enough" global functions on the surface. That is by no means obvious and requires some hard work to be proven. I would look at Forster's Lectures on Riemann Surfaces if you want to see this proven.
By the way, there are some ways to sidestep the Riemann Existence Theorem in some sense but there is as always a conservation of difficulty. If you prove the Kodaira Embedding Theorem as in Huybrechts' Complex Geometry, you can conclude that compact Riemann surfaces are projective and hence algebraic again by Chow's Theorem. The issue is that this assumes results from Hodge Theory which have their own analytic technicalities.
Long story short, these conditions turn out to be equivalent to being algebraic (and you should probably not worry about it further for a while unless it is the sort of thing you find interesting).